Advanced automated fiber placement technologies facilitate the fabrication of variable angle tow (VAT) composites, offering enhanced flexibility for stiffness-tailored lightweight structures. Nevertheless, the temperature-dependency of these materials imposes significant computational constraints on the nonlinear bifurcation analysis of thermally loaded VAT plates featuring complex cutout geometries. In this work, an isogeometric nonlinear analysis framework integrating robust path-following and branch-switching is developed to investigate thermally-induced postbuckling behaviors of temperature-dependent VAT plates with cutouts. The formulations are derived based on Reddy's third-order shear deformation theory, incorporating von Kármán's nonlinearity and temperature-dependent effects. The nonlinear bifurcation equations for thermal loading scenarios are firstly established based on the principle of minimum thermoelastic potential energy and subsequently solved within an isogeometric nonlinear analysis framework. The proposed framework incorporates temperature-dependent effects via a forward difference scheme, enabling robust handling of thermally-induced nonlinear bifurcations through three core capabilities: (i) resolving snap-through/snap-back instabilities with path-following techniques, (ii) locating singular points via pinpointing techniques, and (iii) switching to secondary paths at bifurcation points using branch-switching techniques. The accuracy and effectiveness of the proposed isogeometric nonlinear analysis framework are verified against published benchmark results and FE solutions from ABAQUS. The influences of temperature-dependent material properties on the nonlinear bifurcation behaviors of thermally loaded VAT plates with cutouts are also explored in several numerical examples.